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3.2. Discrete LTI System

x(n)   ->   y(n)

Time invariance:

x(n+m)   ->   y(n+m)

Linearity:

x1(n)  ->   y1(n) => ax1(n) + bx2(n)   ->   ay1(n) + by2(n)
x2(n)  ->   y2(n)

The discrete unit impulse function is defined by

(n) = 1, n = 0
0, n 0.

Let the systems output corresponding to the unit impulse input be h(n), the impulse response of the system,

(n)   ->   h(n).

It can be shown that the output is

y(n) = h(k) x(n-k)

which is a discrete convolution y(n) = h(n) * x(n). As in the analog case, now instead of the Fourier transform we have the Z-transform, we obtain

Y(z) = H(z) X(z),

H(z) = Y(z) .
X(z)

H(z) is called the transfer function of the system. It is the Z-transform of the impulse response h(n),

H(z) = h(k) z-k

The Z-transforms X(z) and Y(z) are defined similarly.

Example 3.2.1: Finding the transfer function

A discrete system is causal if h(n) = 0, when n < 0. Then

y(n) = h(k) x(n-k).

If also the input is causal, x(n) = 0 when n < 0, then

y(n) = h(k) x(n-k).

Example 3.2.2: Digital derivator


Problems: P36
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